Advertisements
Advertisements
प्रश्न
Choose the correct option from the given alternatives:
`"x"^2/"a"^2 - "y"^2/"b"^2 = 1` is a solution of
विकल्प
`("d"^2"y")/"dx"^2 + "yx" + ("dy"/"dx")^2 = 0`
`"xy"*("d"^2"y")/"dx"^2 + "x"("dy"/"dx")^2 - "y" "dy"/"dx" = 0`
`"y" ("d"^2"y")/"dx"^2 + 2 ("dy"/"dx")^2 + "y" = 0`
`"xy" "dy"/"dx" + "y" ("d"^2"y")/"dx"^2 = 0`
Advertisements
उत्तर
`"xy"*("d"^2"y")/"dx"^2 + "x"("dy"/"dx")^2 - "y" "dy"/"dx" = 0`
Hint:
`"x"^2/"a"^2 - "y"^2/"b"^2 = 1` .....(1)
∴ `1/"a"^2 xx "2x" - 1/"b"^2 xx "2y" "dy"/"dx" = 0`
∴ `"x"/"a"^2 - "y"/"b"^2 "dy"/"dx" = 0` ....(2)
and `1/"a"^2 xx 1 - 1/"b"^2 ["y" ("d"^2"y")/"dx"^2 + ("dy"/"dx")^2] = 0` ....(3)
Equations (1), (2) and (3) are consistent
∴ `|("x"^2, - "y"^2, 1),("x", -"y" "dy"/"dx", 0),(1, -["y" ("d"^2"y")/"dx"^2 + ("dy"/"dx")^2], 0)| = 0`
∴ `"xy"*("d"^2"y")/"dx"^2 + "x"("dy"/"dx")^2 - "y" "dy"/"dx" = 0`
APPEARS IN
संबंधित प्रश्न
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
x3 + y3 = 4ax
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
Ax2 + By2 = 1
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
(y - a)2 = 4(x - b)
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
y = c1e2x + c2e5x
Form the differential equation of all parabolas whose axis is the X-axis.
In the following example verify that the given expression is a solution of the corresponding differential equation:
y = `"e"^"ax"; "x" "dy"/"dx" = "y" log "y"`
Solve the following differential equation:
`2"e"^("x + 2y") "dx" - 3"dy" = 0`
Solve the following differential equation:
`"dy"/"dx" = "e"^("x + y") + "x"^2 "e"^"y"`
Reduce the following differential equation to the variable separable form and hence solve:
`"x + y""dy"/"dx" = sec("x"^2 + "y"^2)`
Reduce the following differential equation to the variable separable form and hence solve:
`cos^2 ("x - 2y") = 1 - 2 "dy"/"dx"`
Solve the following differential equation:
(x2 + y2)dx - 2xy dy = 0
Choose the correct option from the given alternatives:
The differential equation of y = `"c"^2 + "c"/"x"` is
Choose the correct option from the given alternatives:
x2 + y2 = a2 is a solution of
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
`"y"^2 = "a"("b - x")("b + x")`
In the following example verify that the given function is a solution of the differential equation.
`"x"^2 + "y"^2 = "r"^2; "x" "dy"/"dx" + "r" sqrt(1 + ("dy"/"dx")^2) = "y"`
In the following example verify that the given function is a solution of the differential equation.
`"y" = "e"^"ax" sin "bx"; ("d"^2"y")/"dx"^2 - 2"a" "dy"/"dx" + ("a"^2 + "b"^2)"y" = 0`
In the following example verify that the given function is a solution of the differential equation.
`"y" = 3 "cos" (log "x") + 4 sin (log "x"); "x"^2 ("d"^2"y")/"dx"^2 + "x" "dy"/"dx" + "y" = 0`
In the following example verify that the given function is a solution of the differential equation.
`"xy" = "ae"^"x" + "be"^-"x" + "x"^2; "x" ("d"^2"y")/"dx"^2 + 2 "dy"/"dx" + "x"^2 = "xy" + 2`
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
y = `sqrt("a" cos (log "x") + "b" sin (log "x"))`
Form the differential equation of the hyperbola whose length of transverse and conjugate axes are half of that of the given hyperbola `"x"^2/16 - "y"^2/36 = "k"`.
Solve the following differential equation:
x dy = (x + y + 1) dx
Solve the following differential equation:
y log y = (log y2 - x) `"dy"/"dx"`
Solve the following differential equation:
`"dx"/"dy" + "8x" = 5"e"^(- 3"y")`
Find the particular solution of the following differential equation:
`("x + 2y"^2) "dy"/"dx" = "y",` when x = 2, y = 1
Find the particular solution of the following differential equation:
`2e ^(x/y) dx + (y - 2xe^(x/y)) dy = 0," When" y (0) = 1`
Select and write the correct alternative from the given option for the question
The solutiion of `("d"y)/("d"x) + x^2/y^2` = 0 is
Find the general solution of `("d"y)/("d"x) = (1 + y^2)/(1 + x^2)`
Form the differential equation of family of standard circle
Find the differential equation from the relation x2 + 4y2 = 4b2
The family of curves y = `e^("a" sin x)`, where a is an arbitrary constant, is represented by the differential equation.
Find the differential equation of the family of circles passing through the origin and having their centres on the x-axis
Choose the correct alternative:
The slope at any point of a curve y = f(x) is given by `("d"y)/("d"x) - 3x^2` and it passes through (-1, 1). Then the equation of the curve is
The rate of disintegration of a radio active element at time t is proportional to its mass, at the time. Then the time during which the original mass of 1.5 gm. Will disintegrate into its mass of 0.5 gm. is proportional to ______.
The elimination of the arbitrary constant m from the equation y = emx gives the differential equation ______.
The differential equation for all the straight lines which are at the distance of 2 units from the origin is ______.
Solve the following differential equation:
`xsin(y/x)dy = [ysin(y/x) - x]dx`
For the curve C: (x2 + y2 – 3) + (x2 – y2 – 1)5 = 0, the value of 3y' – y3 y", at the point (α, α), α < 0, on C, is equal to ______.
If y = (tan–1 x)2 then `(x^2 + 1)^2 (d^2y)/(dx^2) + 2x(x^2 + 1) (dy)/(dx)` = ______.
The differential equation of all parabolas having vertex at the origin and axis along positive Y-axis is ______.
Form the differential equation whose general solution is y = a cos 2x + b sin 2x.
A particle is moving along the X-axis. Its acceleration at time t is proportional to its velocity at that time. Find the differential equation of the motion of the particle.
The differential equation whose solution represents the family \[x^{2}y=4e^{x}+c\], where c is an arbitrary constant, is
